Detectability of a change point depends on the magnitude of the corresponding change,
and selecting a norm for measuring the magnitude of \(\mathbf {d}_j\) 
effectively corresponds to modelling the (cross-sectional) type of a change
for which the proposed change point methodology is particularly tailored.
High-dimensional efficiency of change point tests, as proposed in [1]},
relates to the separation rate between detectable and undetectable changes.
The norms associated with the high-dimensional efficiency of
different methods proposed in the existing literature indeed vary:
Methods designed for detecting sparse changes
(where \(\Vert \mathbf {d}_j \Vert _0 := \sum _{i = 1}^p \mathbb {I}_{\vert d_{ij} \vert > 0} \ll p\) )
such as those proposed in [2]}, [3]}, [4]}, [5]},
the scan statistic of [6]},
are associated with the \(\ell _\infty \) -norm
(sometimes implicitly so via \(\Vert \mathbf {d}_j \Vert /\sqrt{\Vert \mathbf {d}_j \Vert _0}\) ),
while the methods from [7]}, [8]}
and the linear statistic of [6]}
are associated with the \(\ell _2\) -norm,
see Table REF  for a summary which extends Table 1 of [10]}.
<TABLE>